How can I "zero" an object's orientation in blender?
The simplest ways to do this are to either rotate the mesh in Edit Mode or to rotate it in Object mode and then applying the rotation.

So you can either enter Edit mode, by pressing Tab, and then rotating the mesh with R; the reason this works is that, when transforms are applied in edit mode, the object's origin is not affected.

The other possible solution is to press Ctrl + A>Rotation, which will, essentially do the same as above, by rotating the mesh, but not the object, and thus not the origin, and then resseting the origin's roattion.